Clandestine Simulations in Cellular Automata

TL;DR

This paper explores two types of simulation in cellular automata, revealing their contrasting effects on complexity and demonstrating how any CA can be embedded into simpler systems with limited complexity.

Contribution

It introduces and analyzes the contrasting behaviors of factor and sub-automaton simulations, showing how complex CA can be embedded into simpler CA with limited dynamical complexity.

Findings

Factor simulation preserves complexity of attractors and subshifts.

Any CA can be embedded into a CA with simple limit set.

Existence of intrinsically universal CA with simple dynamics.

Abstract

This paper studies two kinds of simulation between cellular automata: simulations based on factor and simulations based on sub-automaton. We show that these two kinds of simulation behave in two opposite ways with respect to the complexity of attractors and factor subshifts. On the one hand, the factor simulation preserves the complexity of limits sets or column factors (the simulator CA must have a higher complexity than the simulated CA). On the other hand, we show that any CA is the sub-automaton of some CA with a simple limit set (NL-recognizable) and the sub-automaton of some CA with a simple column factor (finite type). As a corollary, we get intrinsically universal CA with simple limit sets or simple column factors. Hence we are able to 'hide' the simulation power of any CA under simple dynamical indicators.